Finite Group DatabasebetaFinite Group Databasebeta| order | 1024 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C215 |
| direct product | C23×#2205 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×Q8 | C26 |
| prime radical | C23×2−1+4 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C25 ◁ C24×Q8 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
35777 conjugacy classes of subgroups; only the 4053 normal subgroups are shown.
This group has 35777 conjugacy classes of subgroups — too many to draw. Only its 4053 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 128 |
160 conjugacy classes — view the full character table.
| Fitting subgroup of | C2×#72212285 |
| maximal subgroup of | C2×#72212208, C2×#72212285 |
| maximal normal subgroup of | C2×#72212208, C2×#72212285 |
Past the SmallGroups library coverage — no SmallGroups id.